Fate of Accidental Symmetries of the Relativistic Hydrogen Atom in a Spherical Cavity

TL;DR

This paper investigates how the accidental $SO(4)$ symmetry of the hydrogen atom is affected by relativistic effects and boundary conditions within a spherical cavity, revealing conditions under which degeneracies persist or are lifted.

Contribution

It extends the analysis of accidental symmetries to the relativistic hydrogen atom with general boundary conditions, showing that such symmetries are generally broken in finite volumes.

Findings

Accidental symmetry is partially preserved in non-relativistic case with specific boundary conditions.

Relativistic case generally lifts the accidental symmetry in finite volume.

Some degeneracies remain for particular cavity sizes and boundary parameters.

Abstract

The non-relativistic hydrogen atom enjoys an accidental $SO(4)$ symmetry, that enlarges the rotational $SO(3)$ symmetry, by extending the angular momentum algebra with the Runge-Lenz vector. In the relativistic hydrogen atom the accidental symmetry is partially lifted. Due to the Johnson-Lippmann operator, which commutes with the Dirac Hamiltonian, some degeneracy remains. When the non-relativistic hydrogen atom is put in a spherical cavity of radius $R$ with perfectly reflecting Robin boundary conditions, characterized by a self-adjoint extension parameter $\gamma$, in general the accidental $SO(4)$ symmetry is lifted. However, for $R = (l+1)(l+2) a$ (where $a$ is the Bohr radius and $l$ is the orbital angular momentum) some degeneracy remains when $\gamma = \infty$ or $\gamma = \frac{2}{R}$. In the relativistic case, we consider the most general spherically and parity invariant boundary condition, which is characterized by a self-adjoint extension parameter. In this case, the remnant accidental symmetry is always lifted in a finite volume. We also investigate the accidental symmetry in the context of the Pauli equation, which sheds light on the proper non-relativistic treatment including spin. In that case, again some degeneracy remains for specific values of $R$ and $\gamma$.